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ON  THE  MOTION  OF  PENDULUMS.                         115
in the middle of an experiment with the geometric mean of the initial and final arcs.    I have treated in this way Bessel's experiments, Nos. 1, 2, 3, 4, and 5.    Each of these is in fact a group of six experiments, four with the long pendulum and two with the short, so that the whole  consists  of   20  experiments with the long pendulum, and 10 with the short.    In the case of the long pendulum, the observed value of p regularly fell short of the calculated value, and that by a tolerably constant quantity.    The mean difference amounted to 0*688 line, and the mean error in this quantity to 0'109.    This mean error was not due entirely to errors of observation, or variations in the state of the air, &c., but partly also to slight variations in the initial arc, larger differences usually accompanying larger initial arcs.    The initial arc usually corresponded to p = 39 or 40 lines, and the final to p = 15 or 16 lines.    In the case of the short pendulum, the differences in 8 cases out of 10 had the same sign as before.    The mean difference was 0*025, and the mean error 0*043.    The arcs of oscillation were nearly  the same as before; but inasmuch as the axis of suspension was nearer to the scale than before, the initial value of p was only about 12 or 13 lines, and the final value about 7 lines.    When the results of some of the experiments were laid down on paper, by abscissae taken proportional to the times and ordinates to the logarithms of /j,, it was found that in the case of the long pendulum the line so drawn was decidedly curved, the concavity being turned toward the side of the positive ordinates. The curvature of the line belonging to the short pendulum could hardly be made out, or at least separated from the effects of errors of observation.    The experiments 9, 10, 11, having been treated numerically  in the same way as the experiments 15, led to much the same result.    In the  1C experiments with the ivory sphere and short pendulum contained in the experiments Nos. 12, 13, 14, and 15, the excess of the  calculated over the observed value of fju was more apparent, the mean excess amounting to 0*129.    The reason of this probably was, that the observations with the ivory sphere were made  through   a  somewhat   wider range of arc than those with the brass sphere.
It appears then that at least in the case of the long pendulum a correction is necessary, in order to clear the observed decrease in the arc of oscillation from the effect of that part of the